Gauge symmetries and covariant derivatives
Definition: Gauge symmetries
Symmetries whose parametrization $\alpha(x)$ depends on $x$ are called gauge or local symmetries.
Definition: Global symmetries
Symmetries whose parametrization is a constant $\alpha$ not depending on $x$ are called global symmetries.
A global symmetry implies a conserved current according to Noethers theorem.
We can impose gauge symmetry on a Lagrangian containing a vector field $A_\mu(x)$ as it will guarantee the right amount of degrees of freedom for our Representations of the Poincare group.
If we want to construct an interacting theory with e.g. a scalar field $\phi$ all other terms need to respect the gauge symmetry as well.
For an interacting theory of a vector $A_\mu$ we need at least two addtional fields (thus a complex scalar field) $$\phi(x) = \phi_1(x) + i\phi_2(x)$$ which transforms as $$\phi(x) \to e^{-i\alpha(x)}\phi(x)$$ under gauge transformation in order to be able to build interaction terms with $A$ which are gauge invariant. This requirement effectively reduces the degrees of freedom of the scalar field theory to one physical degree of freedom instead of the two components of the field.
To make the kinetic term $\partial_\mu \phi \partial^\mu \phi$ gauge invariant we introduce the Covariant derivative
Definition: Covariant derivative
The covariant derivative $D_\mu$ is defined as a derivative which leaves the kinetic terms of a theory gauge invariant. For a theory with a vector field $A_\mu$ and a complex scalar $\phi$ it is defined as $$D_\mu \phi = (\partial_\mu + ieA_\mu)\phi,$$ with the charge of $\phi$ as $e$. This term transofrmas as $$D_\mu\phi \to e^{-i\alpha(x)}D_\mu\phi,$$ under gauge transformations
Using this in the case of a massless vector $A_\mu$ and a complex massive scalar leads to scalar QED:
Definition: Scalar QED
Scalar QED is a theory with a massless vector $A_\mu$ and a massive complex scalar $\phi$: $$\mathcal{L} = -\frac{1}{4}F_{\mu\nu}F^{\mu\nu} + (D_\mu\phi)^*(D^\mu\phi) - m^2\phi^*\phi$$ It is gauge invariant under $$\phi(x) \to e^{-i\alpha(x)}\phi(x), \quad A_\mu(x) \to A_\mu(x) + \frac{1}{e}\partial_\mu \alpha(x).$$
Note:
- One can define the gauge transformation more general for an arbitrary charge $Q$ as $e^{Qi\alpha(x)}$
- Above we used $Q=-1$ as for an electron.
# Is gauge invariance physical ?
Gauge invariance is not physical!
We just use it to write down a local lagrangian for a given representation
E.g. we want to embed two degrees of freedom in a local lagrangian (because that is the representation of the lorentz group we have)
Then we can write down a gauge invariant lagrangian for a massless vector field $A_\mu$ and because of gauge invariance + equation of motion this will reduce to 2 degrees of freedom
The key insight is how to pour a given number of d.o.f. into a lagrangian of fields and for that we can use gauge invariance
Origin of Gauge Invariance
Gauge Invariance is imposed when embedding the representations of the Poincare Group into fields. E.g. a vector field $V_\mu(x)$ has four degrees of freedom corresponding to Spin 0 and Spin 1. We can construct a theory for massive spin 1 by choosing the Lagrangian in a way so that only the Spin 1 component is propagated. This is exactly gauge invariance.